A note on rank 1 log extendable isocrystals on simply connected open varieties

Abstract

In 2010 de Jong proposed a p-adic version of Gieseker's conjecture: if π ét 1 (X) = 1, for X smooth connected projective variety, then any isocrystal on X is constant. This was proven by Esnault and Shiho [7],[8] under some additional assumptions. We show that the conjecture holds in the case of a non-proper variety with trivial tame fundamental group and rank 1 log extendable isocrystals.